The Lie groups of 2D/3D spacial rotations, \(SO(2)\) /\(SO(3)\) , and translations, \(SE(2)\) /\(SE(3)\) , fully characterize the primitive geometric motion of rigid body, and are broadly used in robotics [1]}, [2]}, [3]}. Both groups admit matrix representations [4]}. Here, the group composition operator, \(\mathbf {\circ }\) , is the standard matrix multiplication and \(\exp \)  the matrix exponential. We review the properties of these groups in the Supplementary Material. In this work, we assume the bases of \(SO(3)\)  and \(SE(3)\)  are known, which is a common setup in many robotic applications. In the case of unknown basis, it can potentially be learned as well [5]}, which we leave for future investigation. In contrast to [5]}, we utilize known group structures as a prior for learning of dynamical model (REF ) for planning.
